Wolfrich lived in Portugal and Brazil for a total period of $14$ months in order to learn Portuguese. He learned an average of $130$ new words per month when he lived in Portugal and an average of $150$ new words per month when he lived in Brazil. In total, he learned $1920$ new words. How long did Wolfrich live in Portugal, and how long did he live in Brazil? Wolfrich lived in Portugal for
Answer: Let $x$ represent the number of months Wolfrich lived in Portugal and let $y$ represent the number of months he lived in Brazil. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that Wolfrich learned an average of $\textit{130}$ new words per month in Portugal, an average speed of $\textit{150}$ new words per month in Brazil, and a total of $\textit{1920}$ new words in Portugal and Brazil. How can we model this sentence algebraically? The total number of new words Wolfrich learned in Portugal can be modeled by $130x$, and the total number of new words he learned in Brazil can be modeled by $150y$. Since these together add up to $1920$, we get the following equation: $130x+150y=1920$ We are also given that Wolfrich lived in Portugal and Brazil for a total of $\textit{14}$ months. This can be expressed as: $x + y =14$ Now that we have a system of two equations, we can go ahead and solve it! We can now solve the system of equations by the elimination method. Note that the coefficient of $y$ in the first equation, $150$, is exactly $150$ times the coefficient of $y$ in the second equation, $1$. Therefore, we can multiply the second equation by ${-150}$ in order to eliminate $y$. $\begin{aligned} {-150}\cdot x+({-150})\cdot y&={-150}\cdot14\\\\ -150x-150y&=-2100\end{aligned}$ Now we can eliminate $y$ : $\begin{aligned}130x+{150y}&=1920\\\\ {+}\ -150x-{150y}&=-2100\\ \hline\\ -20x+0 &=-180 \end{aligned}$ When we solve the resulting equation, we find that $x =9$, which we can substitute into $x+y=14$ to find that $y=5$. Recall that $x$ denotes the time Wolfrich lived in Portugal and $y$ denotes the time he lived in Brazil. Therefore, Wolfrich lived in Portugal for $\textit{9}$ months and lived in Brazil for $\textit{5}$ months.